Johan U. Backstrom

Quadratic programming algorithms for large-scale model predictive control

Abstract Quadratic programming (QP) methods are an important element in the application of model predictive control (MPC). As larger and more challenging MPC applications are considered, more attention needs to be focused on the construction and tailoring of efficient QP algorithms. In this study, we tailor and apply a new QP method, called QPSchur, to large MPC applications, such as cross directional control problems in paper machines. Written in C++, QPSchur is an object oriented implementation of a novel dual space, Schur complement algorithm. We compare this approach to three widely applied QP algorithms and show that QPSchur is significantly more efficient (up to two orders of magnitude) than the other algorithms. In addition, detailed simulations are considered that demonstrate the importance of the flexible, object oriented construction of QPSchur, along with additional features for constraint handling, warm starts and partial solution.

Dual adaptive control of paper coating

The paper is concerned with the cross directional (CD) control of coat weight on industrial bent blade coaters. The coater is a coupled multivariable process whose gain drifts over time and often switches sign. Current industrial practice is to switch off automatic control when the loop becomes unstable due to gain sign reversal. Because of this, the standard industrial controller is rarely on for more than half of the blade life. The control strategy developed in the paper is based primarily on viewing the coater as a plant with uncertain linear characteristics. An active suboptimal dual controller minimizes a nonlinear performance index designed specifically to reflect the peculiarities of the process. Thus, no heuristic logic is needed. The controller, which takes into consideration the loading and bending limitations imposed by the actuators and the blade, is coupled with an adaptive Kalman filter for parameter estimation. An approximate analytical control law was derived to allow for easy implementation. The proposed strategy was successfully applied to an off-machine bent blade coater. The results of a series of trials show the advantages of including probing in the control signal, and that the adaptive Kalman filter was capable of tracking gain variations. The dual controller yielded substantial quality improvement and was able to control the process throughout the entire blade life. The developed strategy was well accepted by the company.

Approximate steady-state performance prediction of large-scale constrained model predictive control systems

When tuning the parameters of a constrained model predictive controller (MPC), one usually will use closed-loop simulations in order to predict closed-loop performance. Closed-loop simulation can be very time-consuming and inconvenient for large-scale constrained MPC, such as paper machine cross-directional (CD) predictive control. Paper machine CD processes are two-dimensional (2-D) (temporal and spatial) systems with up to 600 inputs and 6000 outputs. It is very important to predict the steady-state values for the closed-loop CD MPC systems during the tuning process, as the variances of these values are used as the control performance indexes in paper making industry. This article proposes to use a direct one-step static optimizer for approximating the closed-loop steady-state performance of constrained CD MPC. The parameters of this static optimizer can be obtained through minimizing the difference of two closed-loop transfer functions. Experiments with industrial data demonstrate that the static optimizer is computationally much more efficient (up to two orders of magnitude) than closed-loop simulation while reliably and accurately predicting the steady-state performance.

Model Predictive Control and Optimization for Papermaking Processes

Papermaking is a large-scale two-dimensional process. It has to be monitored and controlled continuously in order to ensure that the qualities of paper products stay within their specifications. There are two types of control problems involved in papermaking processes: machine directional (MD) control and cross directional (CD) control. Machine direction refers to the direction in which paper sheet travels and cross direction refers to the direction perpendicular to machine direction. The objectives of MD control and CD control are to minimize the variation of the sheet quality measurements in machine direction and cross direction, respectively. This chapter considers the design and applications of model predictive control (MPC) for papermaking MD and CD processes. MPC, also known as moving horizon control (MHC), originated in the late seventies and has developed considerably in the past two decades (Bemporad and Morari 2004; Froisy 1994; Garcia et al. 1998; Morari & Lee 1999; Rawlings 1999; Chu 2006). It can explicitly incorporate the process’ physical constraints in the controller design and formulate the controller design problem into an optimization problem. MPC has become the most widely accepted advanced control scheme in industries. There are over 3000 commercial MPC implementations in different areas, including petro-chemicals, food processing, automotives, aerospace, and pulp and paper (Qin and Badgwell 2000; Qin and Badgwell 2003). Honeywell introduced MPC for MD controls in 1994; this is likely the first time MPC technology was applied to MD controls (Backstrom and Baker, 2008). Increasingly, paper producers are adopting MPC as a standard approach for advanced MD controls. MD control of paper machines requires regulation of a number of quality variables, such as paper dry weight, moisture, ash content, caliper, etc. All of these variables may be coupled to the process manipulated variables (MV’s), including thick stock flow, steam section pressures, filler flow, machine speed, and disturbance variables (DV’s) such as slice lip adjustments, thick stock consistency, broke recycle, and others. Paper machine MD control is truly a multivariable control problem. In addition to regulation of the quality variables during normal operation, a modern advanced control system for a paper machine may be expected to provide dynamic economic optimization on the machine to reduce energy costs and eliminate waste of raw materials. For machines that produce more than one grade of paper, it is desired to have an automatic grade change feature that will create and track controlled variable (CV) and MV

Sensitivity of controller performance indices to model-plant mismatch: An application to paper machine control

The performance of a model-based controller depends inextricably on the quality of the corresponding model. The performance of such a controller is optimal with respect to the model. Therefore any model-plant mismatch can be expected to result in poor performance. The objective of this work is to study the sensitivity of commonly used performance indices to model-plant mismatch. A sensitivity measure is defined and a frequency domain expression for quantifying the sensitivity is derived. It is shown that the sensitivity of the performance indices varies for different types of model mismatch, e.g., gain mismatch, etc. Furthermore, model mismatch has different effects on the performance of the system when it is operating in steady state or in a transitional mode. These differences can reveal the type of plant-model mismatch. The results are illustrated on industrial paper machine data.

Performance Assessment of Cross-Directional Control for Paper Machines

The minimum variance controller has been extensively used as a benchmark in the performance assessment of both univariate and multivariate control loops when time delay is the fundamental performance limitation. In this paper, the spatial and temporal performance limitations in the cross-directional (CD) control of paper machines are analyzed. The idea of minimum variance benchmarking is extended to the CD process based on these performance limitations. Based on an industrial CD controller, a user-specified benchmark, which is more practical and less aggressive, is also proposed. In addition, several related performance indices are proposed for the CD process based on both the minimum variance benchmark and the user-specified benchmark. Illustrative examples from a paper machine simulator and industrial data sets are provided to show the effectiveness of the proposed performance indices.

Detecting model-plant mismatch without external excitation

Any discrepancy between a process and the associated model used in control design will compromise closed-loop performance. In almost all current techniques to detect model-plant mismatch in model-based control systems there must be some sort of external excitation to overcome the effect of unmeasured disturbances on closed-loop signals. In this paper, we propose a novel technique that enables us to detect model-plant mismatch without introducing any external excitation. We show that model-plant mismatch in a closed loop system changes the cross-correlation coefficients between the model prediction error and the process input at certain lags. Indeed, by comparing the correlation between prediction error and input signals in the case of poor performance with that under good performance, one can detect model-plant mismatch. The results are illustrated on paper machine data.

Sensitivity of MIMO controller performance to model-plant mismatch, with applications to paper machine control

Model-based controllers based on incorrect estimates of the true plant behaviour can be expected to perform badly. This work quantifies the performance deterioration for a certain class of MIMO systems. Performance is measured using a Minimum Variance index and a closely related user-specified criterion. Under reasonable conditions, the performance of each output component in a MIMO system can be analysed independently. We define a sensitivity measure that relates system performance to model-plant mismatch, and use it to explore this sensitivity for three realistic types of parametric modelling errors. Next, we suggest a quantitative method that compares a systems actual output to its desired response in a transient setting. The performance of the transient response is demonstrably more sensitive to the model-plant mismatch than the steady state performance. The results are illustrated on industrial paper machine data.

An efficient quadratic programming implementation for cross directional control of large papermaking processes

In this paper, we consider a linearly constrained quadratic programming (QP) problem arising from cross directional control of large papermaking processes. Different from general-purpose QP solvers, we solve the optimization problem by taking advantage of the problem structure and features, such as positive-definiteness of the Hessian matrix, sparsity of the Hessian and constraint matrices. It is implemented based on a dual feasible, active-set algorithm, a Schur complement method and a warm start strategy. The Schur complement is proved to be nonsingular throughout iterations, which makes the solver numerically very reliable. In comparison with the standard Matlab QP solver, the proposed QP solver is much more efficient in the case studies we performed on real industrial papermaking processes.

Robust Tuning of Cross-Directional Model Predictive Controllers for Paper-Making Processes

This paper studies automated tuning of cross-directional model predictive control for industrial paper-making processes under user-specified model parameter uncertainties. Automated parameter tuning algorithms are developed to reduce the variability of the actuator and measurement profiles in the spatial domain and to achieve satisfactory performance in terms of worst case settling times and worst case control signal overshoots in the temporal domain for given parametric uncertainties. Due to decoupling properties of the spatial and temporal frequency components, the controller design and parameter tuning can be realized separately. For the spatial design and parameter tuning in the presence of parametric uncertainties, the undesirable high-frequency components in the actuator profile are suppressed via an appropriate design of the weighting matrix